Finite Group Databasebeta

← #13172218

Subgroups — #13172218

Order 1728.

1146 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723715C211yesmaximal, characteristic
2#11723640C212yesmaximal
3#11723595C211yesmaximal, characteristic
4C3×#63964C212yesmaximal
5C2×C3×Dic32C211yesmaximal, characteristic
6#11723033C212yesmaximal, complemented
7#11723014C212yesmaximal, complemented
8#11722943C212yesmaximal, complemented
9#11722940C212yesmaximal, complemented
10#10594808—32nomaximal
11#10594030—31nomaximal
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C22×C32×Dic3C2212yes—
14C2×#6293C2212yes—
15C2×C3×#627C2212yes—
16#88864C2211yescharacteristic
17C3×#3757C2212yes—
18C32×#217C2211yescharacteristic
19#88815C2212yes—
20#88813C2211yescharacteristic
21#88813C2211yescharacteristic
22#88807C2212yes—
23#88807C2212yes—
24#88806C2212yes—
25Dic3⋊(C3⋊Dic3)C2212yes—
26Dic3×C3⋊Dic3C2212yes—
27C3×#3724C2212yes—
28C3×Dic3⋊Dic3C2212yes—
29C3×Dic3⋊Dic3C2212yes—
30C3×Dic32C2211yescharacteristic
31C3×Dic32C2211yescharacteristic
32C3×Dic32C2212yes—
33#64133—32no—
34#64086—31no—
35C3×#924—32no—
36C2×C3×C4×Dic3—32no—
37C3×#878—32no—
38#63964—32no—
39#63964—32no—
40#63964—62no—
41#63964D312yes—
42#63957—31no—
43#63955—32no—
44#63955—32no—
45#63955—61no—
46C2×Dic32D311yescharacteristic
47#63918—32no—
48#63867—32no—
49#63867—32no—
50#63866—32no—
51#63847—32no—
52#63844—32no—
53#63843—32no—
54Dic3×Dic6—32no—
55#63839—32no—
56#63838—32no—
57#63837—32no—
58#63837—32no—
59C23×C33C2311yescharacteristic
60C2×C3⋊(C3⋊Dic3)C2312yes—
61C2×C3×C3⋊Dic3Q814yescomplemented
62C2×C3×C3⋊Dic3C2312yes—
63C2×C3×C3⋊Dic3C2312yes—
64C2×C3×C3⋊Dic3C2311yescharacteristic
65C2×C3×C3⋊Dic3C2311yescharacteristic
66C2×C32×Dic3—24no—
67C2×C32×Dic3C2312yes—
68C2×C32×Dic3C2312yes—
69C2×C32×Dic3C2312yes—
70#6293—24no—
71C3×#627—24no—
72#5622—92no—
73#5470—91no—
74C22×C32×C4—32no—
75C22×C3⋊Dic3C2×D311yescharacteristic
76C2×#634—32no—
77C22×C3×Dic3—22no—
78C22×C3×Dic3—32no—
79C22×C3×Dic3—61no—
80C22×C3×Dic3C2×D312yes—
81C22×C3×Dic3C2×D312yes—
82C2×C3×Dic6—32no—
83C2×C3×Dic6—32no—
84C2×#627—32no—
85C2×#627—32no—
86C2×#627—62no—
87C2×#627C2×D312yes—
88C32×C22⋊C4—31no—
89#3757—31no—
90#3757—32no—
91#3757—61no—
92#3757—62no—
93#3757—121no—
94#3757C2×D312yes—
95C4⋊(C3⋊Dic3)—32no—
96#3750—32no—
97#3750—32no—
98#3750—32no—
99C4×C3⋊Dic3—32no—
100C3×#217—21no—
101C3×#217—32no—
102C3×#217—32no—
103C3×#217—62no—
104C3×#217C2×D312yes—
105C3×C4⋊Dic3—32no—
106C3×C4⋊Dic3—32no—
107C3×Dic3⋊C4—32no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4—32no—
111C3×C4×Dic3—32no—
112C3×C4×Dic3—32no—
113C3×C4×Dic3—32no—
114C3×C4×Dic3—32no—
115#3724—32no—
116#3724—32no—
117#3724—32no—
118#3724—62no—
119#3724C2×D312yes—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—32no—
125Dic3⋊Dic3—32no—
126Dic3⋊Dic3—32no—
127Dic3⋊Dic3—32no—
128Dic3⋊Dic3—62no—
129Dic3⋊Dic3—62no—
130Dic3⋊Dic3—62no—
131Dic3⋊Dic3C2×D312yes—
132Dic3⋊Dic3C2×D312yes—
133Dic32—32no—
134Dic32—32no—
135Dic32—31no—
136Dic32—31no—
137Dic32—32no—
138Dic32—32no—
139Dic32—62no—
140Dic32C2×D312yes—
141Dic32C2×D311yescharacteristic
142Dic32C2×D311yescharacteristic
143C22×C33—22no—
144C22×C33C2×Q812yes—
145C22×C33C2411yescharacteristic
146C3⋊(C3⋊Dic3)—22no—
147C3×C3⋊Dic3—22no—
148C3×C3⋊Dic3—22no—
149C3×C3⋊Dic3C2×Q812yes—
150C3×C3⋊Dic3C2×Q812yes—
151C32×Dic3—22no—
152C32×Dic3—22no—
153C32×Dic3#4114yes—
154C3×C22⋊Q8—92no—
155C2×C3×C42—91no—
156Q8×Dic3—92no—
157#944—92no—
158#926—91no—
159#926—91no—
160#924—32no—
161#924—92no—
162#924—92no—
163#924—182no—
164C2×C4×Dic3—32no—
165#889—92no—
166#889—92no—
167#888—92no—
168#887—92no—
169#887—92no—
170#887—92no—
171#878—32no—
172#878—92no—
173#877—92no—
174#877—92no—
175#877—181no—
176#870—92no—
177#869—92no—
178C4×Dic6—92no—
179C4×Dic6—92no—
180C23×C32—21no—
181C23×C32—22no—
182C23×C32—41no—
183C23×C32C22×D311yescharacteristic
184C23×C32C22×D312yes—
185C2×C32×C4—32no—
186C2×C32×C4—32no—
187C2×C32×C4—32no—
188C2×C32×C4—64no—
189C2×C3⋊Dic3—32no—
190C2×C3⋊Dic3—32no—
191C2×C3⋊Dic3—32no—
192C2×C3⋊Dic3—62no—
193C2×C3⋊Dic3—62no—
194C2×C3⋊Dic3—62no—
195C2×C3⋊Dic3—122no—
196C2×C3⋊Dic3Dic614yescomplemented
197C2×C3⋊Dic3C22×D312yes—
198C2×C3⋊Dic3C22×D312yes—
199C2×C3⋊Dic3C22×D311yescharacteristic
200C2×C3⋊Dic3C22×D311yescharacteristic
201#634—64no—
202C2×C3×Dic3—24no—
203C2×C3×Dic3—24no—
204C2×C3×Dic3—22no—
205C2×C3×Dic3—22no—
206C2×C3×Dic3—22no—
207C2×C3×Dic3—38no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—44no—
215C2×C3×Dic3—61no—
216C2×C3×Dic3—61no—
217C2×C3×Dic3—64no—
218C2×C3×Dic3—62no—
219C2×C3×Dic3—62no—
220C2×C3×Dic3C22×D312yes—
221C2×C3×Dic3C22×D312yes—
222C2×C3×Dic3C22×D312yes—
223C2×C3×Dic3C22×D312yes—
224C2×C3×Dic3C22×D312yes—
225C2×C3×Dic3C22×D312yes—
226C3×Dic6—64no—
227C3×Dic6—64no—
228#627—24no—
229#627—64no—
230#627—64no—
231#627—124no—
232#533—271noSylow
233C2×C33—22no—
234C2×C33C22×Q811yescharacteristic
235C2×C33C2×#4112yes—
236C2×C3×Q8—92no—
237C22×C3×C4—32no—
238C22×C3×C4—32no—
239C22×C3×C4—62no—
240C22×C3×C4—91no—
241C22×Dic3—32no—
242C22×Dic3—61no—
243C22×Dic3D3212yes—
244C2×Dic6—32no—
245C2×Dic6—32no—
246C2×Dic6—92no—
247C2×Dic6—92no—
248C2×Dic6—92no—
249C2×Dic6—182no—
250C3×C4⋊C4—92no—
251C3×C4⋊C4—92no—
252C3×C4⋊C4—92no—
253C3×C22⋊C4—32no—
254C3×C22⋊C4—61no—
255C3×C22⋊C4—92no—
256C3×C42—92no—
257C3×C42—91no—
258C3×C42—91no—
259#217—32no—
260#217—32no—
261#217—62no—
262#217—92no—
263#217—91no—
264#217—181no—
265#217—182no—
266#217—361no—
267#217D3211yescharacteristic
268C4⋊Dic3—32no—
269C4⋊Dic3—32no—
270C4⋊Dic3—92no—
271C4⋊Dic3—92no—
272C4⋊Dic3—92no—
273C4⋊Dic3—92no—
274C4⋊Dic3—92no—
275C4⋊Dic3—182no—
276Dic3⋊C4—32no—
277Dic3⋊C4—32no—
278Dic3⋊C4—32no—
279Dic3⋊C4—32no—
280Dic3⋊C4—92no—
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4—182no—
291Dic3⋊C4—182no—
292Dic3⋊C4—182no—
293C4×Dic3—32no—
294C4×Dic3—32no—
295C4×Dic3—32no—
296C4×Dic3—32no—
297C4×Dic3—91no—
298C4×Dic3—91no—
299C4×Dic3—92no—
300C4×Dic3—92no—
301C4×Dic3—92no—
302C4×Dic3—92no—
303C4×Dic3—92no—
304C4×Dic3—182no—
305C22×C32—22no—
306C22×C32—22no—
307C22×C32—22no—
308C22×C32—22no—
309C22×C32—21no—
310C22×C32—24no—
311C22×C32—22no—
312C22×C32—41no—
313C22×C32—42no—
314C22×C32—42no—
315C22×C32—42no—
316C22×C32—42no—
317C22×C32—82no—
318C22×C32C2×Dic612yes—
319C22×C32D3×Q814yes—
320C22×C32C23×D311yescharacteristic
321C22×C32C23×D312yes—
322C32×C4—34no—
323C32×C4—62no—
324C32×C4—62no—
325C3⋊Dic3—22no—
326C3⋊Dic3—22no—
327C3⋊Dic3—62no—
328C3⋊Dic3—62no—
329C3⋊Dic3—62no—
330C3⋊Dic3—122no—
331C3⋊Dic3—122no—
332C3⋊Dic3—122no—
333C3⋊Dic3—242no—
334C3⋊Dic3C2×Dic612yes—
335C3⋊Dic3C2×Dic612yes—
336C3×Dic3—22no—
337C3×Dic3—22no—
338C3×Dic3—22no—
339C3×Dic3—22no—
340C3×Dic3—24no—
341C3×Dic3—34no—
342C3×Dic3—34no—
343C3×Dic3—42no—
344C3×Dic3—42no—
345C3×Dic3—62no—
346C3×Dic3—62no—
347C3×Dic3—62no—
348C3×Dic3—62no—
349C3×Dic3—62no—
350C3×Dic3—62no—
351C3×Dic3—122no—
352C3×Dic3—122no—
353C3×Dic3#23514yes—
354C3×Dic3#23714yes—
355C4⋊Q8—272no—
356#125—272no—
357C22⋊Q8—92no—
358C22⋊Q8—272no—
359C4×Q8—272no—
360C4×Q8—272no—
361#117—271no—
362#117—271no—
363C2×C42—91no—
364C33#53311yescharacteristic, complemented, Sylow
365C23×C3—22no—
366C23×C3—21no—
367C23×C3—41no—
368C23×C3C2×D3212yes—
369C23×C3C2×D3211yescharacteristic
370C3×Q8—184no—
371C2×C3×C4—32no—
372C2×C3×C4—32no—
373C2×C3×C4—32no—
374C2×C3×C4—32no—
375C2×C3×C4—32no—
376C2×C3×C4—32no—
377C2×C3×C4—62no—
378C2×C3×C4—62no—
379C2×C3×C4—62no—
380C2×C3×C4—64no—
381C2×C3×C4—64no—
382C2×C3×C4—91no—
383C2×C3×C4—91no—
384C2×C3×C4—94no—
385C2×C3×C4—92no—
386C2×C3×C4—92no—
387C2×C3×C4—124no—
388C2×Dic3—24no—
389C2×Dic3—32no—
390C2×Dic3—32no—
391C2×Dic3—38no—
392C2×Dic3—32no—
393C2×Dic3—32no—
394C2×Dic3—32no—
395C2×Dic3—32no—
396C2×Dic3—64no—
397C2×Dic3—61no—
398C2×Dic3—61no—
399C2×Dic3—62no—
400C2×Dic3—62no—
401C2×Dic3—92no—
402C2×Dic3—92no—
403C2×Dic3—92no—
404C2×Dic3—182no—
405C2×Dic3—182no—
406C2×Dic3—182no—
407C2×Dic3—362no—
408C2×Dic3C2×D3212yes—
409C2×Dic3C2×D3212yes—
410C2×Dic3C2×D3212yes—
411Dic6—64no—
412Dic6—64no—
413Dic6—184no—
414Dic6—184no—
415Dic6—184no—
416Dic6—364no—
417C2×C32—22no—
418C2×C32—24no—
419C2×C32—22no—
420C2×C32—21no—
421C2×C32—22no—
422C2×C32—22no—
423C2×C32—22no—
424C2×C32—41no—
425C2×C32—42no—
426C2×C32—42no—
427C2×C32—44no—
428C2×C32—82no—
429C2×C32C22×Dic611yescharacteristic
430C2×C32C2×#23512yes—
431C2×C32C2×#23712yes—
432C2×C32C2×D3×Q812yes—
433C2×C32D3×#4112yes—
434C2×Q8—92no—
435C2×Q8—272no—
436C22×C4—32no—
437C22×C4—91no—
438C4⋊C4—92no—
439C4⋊C4—92no—
440C4⋊C4—92no—
441C4⋊C4—272no—
442C4⋊C4—272no—
443C4⋊C4—272no—
444C4⋊C4—272no—
445C4⋊C4—272no—
446C22⋊C4—31no—
447C22⋊C4—92no—
448C22⋊C4—271no—
449C42—92no—
450C42—91no—
451C42—91no—
452C42—272no—
453C42—271no—
454C42—271no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—21no—
460C22×C3—22no—
461C22×C3—24no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—41no—
466C22×C3—42no—
467C22×C3—82no—
468C22×C3D3×Dic614yes—
469C22×C3#379712yes—
470C22×C3C22×D3212yes—
471C22×C3C22×D3211yescharacteristic
472C3×C4—34no—
473C3×C4—34no—
474C3×C4—64no—
475C3×C4—62no—
476C3×C4—62no—
477C3×C4—62no—
478C3×C4—62no—
479C3×C4—92no—
480C3×C4—92no—
481C3×C4—122no—
482C3×C4—122no—
483C3×C4—182no—
484C3×C4—182no—
485Dic3—22no—
486Dic3—22no—
487Dic3—34no—
488Dic3—34no—
489Dic3—62no—
490Dic3—62no—
491Dic3—62no—
492Dic3—62no—
493Dic3—62no—
494Dic3—62no—
495Dic3—122no—
496Dic3—122no—
497Dic3—182no—
498Dic3—182no—
499Dic3—182no—
500Dic3—362no—
501Dic3—362no—
502Dic3—362no—
503Dic3—722no—
504Dic3#380414yes—
505C32—21no—
506C32—22no—
507C32—41no—
508C32#547011yescharacteristic, complemented
509C32#562212yescomplemented
510C23D3311yescharacteristic
511Q8—184no—
512Q8—544no—
513C2×C4—32no—
514C2×C4—32no—
515C2×C4—32no—
516C2×C4—64no—
517C2×C4—91no—
518C2×C4—91no—
519C2×C4—94no—
520C2×C4—92no—
521C2×C4—92no—
522C2×C4—272no—
523C2×C3—22no—
524C2×C3—24no—
525C2×C3—22no—
526C2×C3—22no—
527C2×C3—22no—
528C2×C3—21no—
529C2×C3—22no—
530C2×C3—42no—
531C2×C3—44no—
532C2×C3—42no—
533C2×C3—41no—
534C2×C3—82no—
535C2×C3C2×D3×Dic612yes—
536C2×C3C2×#379711yescharacteristic
537C2×C3D3×#23512yes—
538C2×C3D3×#23712yes—
539C2×C3C2×#380412yes—
540C22—22no—
541C22#8895712yes—
542C22C2×D3311yescharacteristic
543C4—34no—
544C4—62no—
545C4—62no—
546C4—92no—
547C4—92no—
548C4—182no—
549C4—182no—
550C4—542no—
551C3—21no—
552C3—22no—
553C3—41no—
554C3#1059403011yescharacteristic, complemented
555C3#1059480812yescomplemented
556C2—22no—
557C2D3×#380412yes—
558C2C2×#8895711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
Fitting, prime radicalC23×C33C23
hypercenterC23D33
hypocenterC33#533
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. #11723033C2semidirect
7. #11723014C2semidirect
8. #11722943C2semidirect
9. #11722940C2semidirect
61. C2×C3×C3⋊Dic3Q8semidirect
196. C2×C3⋊Dic3Dic6semidirect
364. C33#533semidirect
508. C32#5470semidirect
509. C32#5622semidirect
554. C3#10594030semidirect
555. C3#10594808semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117237152
#117236402
#117235952
C3×#639642
C2×C3×Dic322
#117230332
#117230142
#117229432
#117229402
#105948083
#105940303